Frontiers in Water Resource Economics (Natural Resource Management and Policy, 29) 0387300554, 9780387300559

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FRONTIERS IN WATER RESOURCE ECONOMICS

NATURAL RESOURCE MANAGEMENT AND POLICY Editors: Ariel Dinar Rural Development Department The World Bank 1818 H Street, NW Washington, DC 20433

David Zilberman Dept. of Agricultural and Resource Economics Univ. of California, Berkeley Berkeley, CA 94720

EDITORIAL STATEMENT There is a growing awareness to the role that natural resources such as water, land, forests and environmental amenities play in our lives. There are many competing uses for natural resources, and society is challenged to manage them for improving social well being. Furthermore, there may be dire consequences to natural resources mismanagement. Renewable resources such as water, land and the environment are linked, and decisions made with regard to one may affect the others. Policy and management of natural resources now require interdisciplinary approach including natural and social sciences to correctly address our society preferences. This series provides a collection of works containing most recent findings on economics, management and policy of renewable biological resources such as water, land, crop protection, sustainable agriculture, technology, and environmental health. It incorporates modem thinking and techniques of economics and management. Books in this series will incorporate knowledge and models of natural phenomena with economics and managerial decision frameworks to assess alternative options for managing natural resources and environment.

The

Series

Editors

Recently Published Books in the Series Haddadin, Munther J. Diplomacy on the Jordan: International Conflict and Negotiated Resolution Renzetti, Steven The Economics of Water Demands Just, Richard E. and Pope, Rulon D. A Comprehensive Assessment of the Role of Risk in U,S. Agriculture Dinar, Ariel and Zilberman, David Economics of Water Resources: The Contributions of Dan Yaron Unver,.I.H. Olcay, Gupta, Rajiv K. IAS, and Kibaroglu, Ay^egiil Water Development and Poverty Reduction d'Estree, Tamra Pearson and Colby, Bonnie G. Braving the Currents: Evaluating Environmental Conflict Resolution in the River Basins of the American West Cooper, Joseph, Lipper, Leslie Marie and Zilberman, David Agricultural Biodiversity and Biotechnology in Economic Development

FRONTIERS IN WATER RESOURCE ECONOMICS

Edited by

RENAN-ULRICH GOETZ University of GIrona

DOLORS BERGA University of Girona

Springer

Library of Congress Control Number: 2005935082

ISBN:10: 0-387-30055-4 ISBN-13: 978-0387-30055-9

e-ISBN-10: 0-387-30056-2 e-ISBN-13: 978-0387-30056-6

Printed on acid-free paper. © 2006 Springer Science+Business Media, Inc. All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, Inc., 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed in the United States of America. 9 8 7 6 5 4 3 2 1 springeronline.com

Contents

List of Figures List of Tables Contributing Authors Acknowledgments Introduction 1 Application of stochastic cooperative games in water resources Ariel Dinar, Stefano Moretti, Fioravante Patrone and Stefano Zara

vii ix xi xv xvii 1

2 Incentives and the search for unknown resources such as water Jean-Jacques Lajfont and Frangois Salanie

21

3 Risk aversion and gains from water trading under uncertain water availability Javier Calatrava Leyva and Alberto Garrido

43

4 Dynamic uncertainty and the pricing of natural monopolies: the case of urban water management Ephra'im Clark and Gerard Mondello 5 Price risk and the diffusion of conservation technology Georgina Moreno and David Sanding 6 Optimal management of groundwater over space and time Nicholas Brozovic, David Sanding and David Zilberman 7 Nonpoint source pollution in a spatial intertemporal context - a deposit refund approach Renan-Ulrich Goetz, Dolors Berga and Angels Xabadia

69

89

109

137

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FRONTIERS IN WATER RESOURCE ECONOMICS

8 Transboundary water management along the U.S.-Mexico border Linda Fernandez 9 Irrigation, water quality and water rights in the Murray DarHng Basin, Australia Stephen Beare and Anna Heaney

153

177

10 Economic analysis of green payments to protect water quality Madhu Khanna and Richard Famsworth

199

11 Conjunctive use of surface and groundwater with quality considerations Catarina Roseta-Palma

225

12

The impact of recovering irrigation water losses on the choice of irrigation technology with heterogeneous land quality and different crops Petra J.G.J. Hellegers 13 Precision farming in cotton David Zilberman, Daniel Cohen-Vogel and Jeanne Reeves

241

255

List of Figures

1.1 1.2 1.3 1.4 2.1 2.2 2.3 2.4 2.5 3.1 3.2 3.3 3.4

6.1 6.2 6.3 6.4 6.5 6.6 9.1 9.2

Cost function of water treatment 6 An intermediate situation between interpretation (a) and (b) 7 Cost of different technologies ti and ^2 for water treatment as a function of amount of water to be treated 14 Consequences (in terms of cost allocations between player 1 and 2) of actions ti and ^2 in each possible state 15 Typical profile of quantity allocated to the ^-agent 26 Impact of A^ on the Principal's payoff 28 Profile of optimal enforcement levels 31 Quantity profiles for the quadratic example 31 Profile of quantities in a continuous type case 39 Excess water demand and supply function under uncertainty 51 Decision tree for the uncertainty problem 53 Iterative modehng process to characterize water price uncertainty 56 Water market profits for each assumption and uncertainty scenario (percentages of profit in absence of uncertainty, model CERT=100) 58 Drawdown from multiple wells in an aquifer 111 Drawdown from multiple wells in a single-cell aquifer 113 Per-period weighting function through time for a steadystate aquifer (distance 1000 feet) 121 Per-period weighting function through time for a steadystate aquifer (distance 25000 feet) 122 Total relative impacts of pumping as a function of distance (high storativity and low transmissivity) 123 Total relative impacts of pumping as a function of distance (low storativity and high transmissivity) 124 Major irrigation areas in the southern Murray Darling Basin 179 Salt concentration of the Murray River at tributary confluences, 2000 182

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FRONTIERS IN WATER RESOURCE ECONOMICS 9.3 9.4 9.5 10.1 11.1 11.2 12.1

13.1

Weighting function for contribution of past recharge to discharge External benefits and internal costs per megalitre of a 20 GL reduction in water allocations External benefits and costs per megalitre of a 5 per cent increase in irrigation efficiency Effects of alternative decision rules for green payments on land targeted for retirement Possible phase diagram for WGG > 0 Phase diagram for ^ > 0 Quasi-rents as a function of land quality for a traditional i = 1 and modem technology i ~ 2, and for two different crops j = 1, 2 under the modem technology (dashed fine) Von Liebig production function

188 192 194 203 232 236

247 266

List of Tables

3.1 3.2 3.3 3.4

3.5

4.1 5.1 5.2 5.3 5.4 5.5 5.6 9.1 9.2 13.1

Farm types considered Scenarios of water uncertainty within a season Models formulation Percentage of total water available in the irrigation district exchanged in the market and water price for each level of water allotment (all assumptions). Scarcity scenario 1 Water market profits for each farm type and model assumption (percentages of profit in absence of uncertainty). Supply uncertainty scenario 1 Parameters information Elasticities of utilization for selected industries Summary statistics for continuous variables Summary statistics for discrete variables Ordered probit estimation results Predicted probability of technology adoption for annual and permanent crops Average elasticities for field characteristics Major irrigation areas in the southern Murray Darling Basin Summary data for irrigation areas in the southern Murray Darling Basin, AustraUa The potential of precision farming in the irrigation of cotton

51 52 54

57

57 83 98 100 100 102 102 103 179 189 271

Contributing Authors

Stephen Beare, Chief Economist, Australian Bureau of Agricultural and Resource Economics, Canberra, Australia Dolors Berga, Associate Professor, Departament d'Economia, Universitat de Girona, Girona, Spain Nicholas Brozovic, Assistant Professor, Department of Agricultural and Consumer Economics, University of Illinois at Urbana-Champaign, Urbana, Illinois, U.S.A. Javier Calatrava Leyva, Associate Professor, Departamento de Economia de la Empresa, Universidad Politecnica de Cartagena, Cartagena, Spain Ephraim Clark, Professor of Finance, Department of Finance, Middlesex University, London, UK Daniel Cohen-Vogel, Senior Policy Analyst, Florida Legislature, Office of Program Policy Analysis and Government Accountability, Tallahassee, Florida, U.S.A. Ariel Dinar, Lead Economist, Department of Agriculture and Rural Development, World Bank, Washington DC, U.S.A. Richard Farnsworth, Associate Professor, Department of Forestry and Natural Resources, Purdue University, West Lafayette, Indiana, U.S.A. Linda Fernandez, Associate Professor, Department of Environmental Sciences, University of Cahfomia at Riverside, Riverside, California, U.S.A.

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FRONTIERS IN WATER RESOURCE ECONOMICS

Alberto Garrido, Associate Professor, Department of Agricultural and Resource Economics, Universidad Politecnica de Madrid, Madrid, Spain Renan-Ulrich Goetz, Associate Professor, Departament of Economics, Universitat de Girona, Girona, Spain Anna Heaney, Manager, Department of Land & Water Economics; Natural Resources Branch, Australian Bureau of Agricultural and Resource Economics, Canberra, Australia Petra J.G.J. Hellegers, Senior Researcher, Department of Management of Natural Resources, Agricultural Economics Research Institute (LEI) (Wageningen University and Research Institute), Wageningen, The Netherlands Madhu Khanna, Professor, Department of Agricultural and Consumer Economics, University of Illinois at Urbana-Champaign, Urbana, Illinois, U.S.A. Jean-Jacques Laffont^, Professor, Universite de Toulouse, Toulouse, and Centre National de la Recherche Scientifique, France; and University of Southern California, U.S.A. Gerard Mondello, Researcher, Groupe de Recherche en Droit, Economic, Gestion; Centre National de la Recherche Scientifique, Sophia AntipoUs, France Georgina A. Moreno, Assistant Professor, Department of Economics, Scripps College, Claremont, California, U.S.A. Stefano Moretti, Ph.D. student. Department of Mathematics, University of Genoa and Unit of Molecular Epidemiology, National Cancer Research Institute of Genoa, Genoa, Italy Fioravante Patrone, Full Professor, Dipartimento di Ingegneria della Produzione, Termoenergetica e Modelli Matematici, University of Genoa, Genoa, Italy Jeanne Reeves, Director, Agricultural Research, Cotton Incorporated, Gary, North Carolina, U.S.A.

Contributing Authors

xiii

Catarina Roseta-Palma, Assistant Professor, Dinamia e Departamento de Economia, Instituto Superior de Ciencias do Trabalho e da Empresa, Lisbon, Portugal Frangois Salanie, Charge de Recherche, Institut National de la Recherche Agronomique, Laboratoire d'Economic des Ressources Naturelles, Universite de Toulouse, Toulouse, France. David Sunding, Professor, Department of Agricultural and Resource Economics, University of California at Berkeley, Berkeley, California, U.S.A. Angels Xabadia, Assistant Professor, Departamento de Analisis Economico, Universidad de Zaragoza, Zaragoza, Spain Stefano Zara, Collaborator, University of Genova, Iglesias (Cagliari), Italy David Zilberman, Professor and Robinson Chair, Department of Agricultural and Resource Economics, Director, Giannini Foundation of Agricultural Economics, University of California at Berkeley, Berkeley, California, U.S.A.

Acknowledgments The editors of this book would Uke to thank the company CESPA Ingenieria Urbana and the Institute of the Environment of the University of Girona for their financial support of the congress (7*^ Conference of the International Water and Resource Economics Consortium and 4^^ Seminar on Environmental and Resource Economics) that gave rise to a large number of papers collected in this book. We would also like to extend our thanks to Angels Xabadia who unified the layout of the different chapters in a timely and excellent manner. Finally our thanks go to the anonymous reviewers for their comments and suggestions that helped to improve the readability of the text and the presentation of complex issues.

Introduction

Most of the books published previously in the field of water resource economics focus on particular aspects of water economics such as institutions, pricing or water markets, but none of them have given particular attention to methodological questions. However, the applied methodology within economic research has made some remarkable advances over the last 10-20 years. Some of these advances are of particular interest to the field of water economics. Therefore, we think that a book that focusing on methodological advances within the field of water resource economics and showing how these advances can be applied in economic analysis of water issues makes a nice complement to the existing literature in this field. We identified five areas where we consider the methodological advances to be of particular importance: 1) asymmetric information and game theory, 2) uncertainty, 3) space, 4) water quality and 5) production and technology adoption. The selected papers for the book fall entirely within these categories. The book ''Frontiers in Water Resource Economics" draws to a great extent on papers which were presented at the 7^^ Conference of the International Water and Resource Economics Consortium, June 3-5,2001 held in Girona, Catalonia, Spain, This conference was jointly organized with the 4^^ Conference of Environmental and Resource Economics by the Department of Economics, University of Girona. The chapters in this book have been selected, in part, from the above mentioned conferences and solicited, in part, from distinguished authors in the field. The composition of the book is based on the five areas identified above. In what follows we briefly present the different chapters for each area. Asymmetric information and game theory (Chapters 1-2) The paper by Dinar, Moretti, Patrone, and Zara, "Application of Stochastic Cooperative Game Theory in Water Resources", applies concepts of stochastic cooperative games to water resource problems. Water resource projects are subject to stochastic water supply patterns, which affect their performance, sustainability and the stability of any use arrangement among participants. Tradi-

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FRONTIERS IN WATER RESOURCE ECONOMICS

tionally, cooperative game theory has been applied to a variety of water resource problems assuming a deterministic pattern of supply. By means of different examples the authors show how incorporation of stochastic considerations may change the solution, depending on the attitude toward risk aversion of the potential users and on the nature of the cost function of the joint water project. Another chapter within this area is by Laffont and Salanie, titled "Incentives and the Search for Unknown Resources such as Water". This work uses the principal-agent methodology to analyze the problems in contracting agents to invest in the search for a natural resource such as water, when these agents are informed about the size of their discoveries but the other parties are not. The chapter is related to three strands of the literature: the principal-agent models with type-dependent status quo payoffs, law and economics literature and the mechanism design literature about the manipulation of endowments. A remarkable result is that limited liability constraints and enforcement problems result in unusual distortions compared to the first best as the principal tries to decrease the agent's effort. Uncertainty (Chapters 3 - 5 ) : The chapter by Calatrava Leyva and Garrido, "Risk Aversion and Gains from Water Trading under Uncertain Water Availability", analyzes on theoretical and empirical grounds the impact of risk-averse behavior and uncertainty of water supply on the efficiency gains from spot water markets. It examines the role that uncertainty and risk aversion play in market participants' decisions and their impact on gains from trade. Irrigators' behavior is simulated using a two-stage discrete stochastic programming model, while market exchanges and equilibrium are computed using a price endogenous spatial equilibrium model. Participation in the water market is modeled as a tactical response, and a utility-efficient modeling approach is used to account for risk aversion. The empirical application is performed simulating a local water market in an irrigation district of the Guadalquivir Valley (Southern Spain). Results show that gains from trade diminish as uncertainty and risk-aversion levels increase. The chapter by Clark and Mondello, "Dynamic Uncertainty and the Pricing of Natural Monopolies: The Case of Urban Water Management", presents a model based on techniques utilized in real option theory. The model can be used as a pricing program that gives a fair deal to both the regulator and the delegated firm. Their pricing program goes beyond marginal cost and non-linear pricing by introducing time and risk. Importantly, it provides a solution to the price that the delegated firm should pay for the right to exploit the monopoly. It also includes the relatively recent problem of technological monopolies that make it possible for the private water firms to extract supplementary economic rents. The fair deal in their pricing program eliminates economic rents while rewarding the risk taken on by the firm. It also guarantees that the regulator

INTRODUCTION

xix

receives the full value of the monopoly that it is ceding to the firm. In this sense the fair deal that they propose is an optimum. The final chapter in this area by Moreno and Sunding, "Price Risk and the Diffusion of Conservation Technology", explores the diffusion pattern of conservation technology in irrigated agriculture. The chapter examines the influence of factor price risk on factor-use efficiency through the adoption of conservation technology. Conceptual results indicate that a mean-preserving increase in factor price risk has an ambiguous effect on returns to investment in conservation technology, but should be related to the own-price elasticity of input utilization. Theoretical results are tested by estimating an ordered probit model of technology choice using a unique data set from the water industry. Estimation results are consistent with the conceptual model, and also indicate that price risk can have a large influence on the adoption decision. Space (Chapters 6 -8): The chapter by Brozovic, Sunding, and Zilberman, "Optimal Management of Groundwater over Space and Time", presents a model for the optimal management over space and time of a groundwater resource with multiple users. Their model extends the existing literature, incorporating both space and time and using the hydraulic response equations that govern the behavior of groundwater. In their analysis, the authors emphasize how physical parameters of the groundwater system affect the spatial and temporal distribution of extraction. A discussion of the optimality conditions from their model emphasizes how the results differ from existing studies and the policy implications of these differences. In particular, they show that some aquifers are more akin to private property than common property, and that there may be significant lagged effects from pumping. The chapter by Goetz, Berga, and Xabadia, "Nonpoint Source Pollution in a Spatial Intertemporal Context - a Deposit Refund Approach", shows that the incorporation of space allows a more precise relationship to be established between a contaminating byproduct and the emissions that reach the final receptor. However, the presence of asymmetric information impedes the implementation of the first-best policy. As a partial solution to their problem they propose a site specific deposit refund for the contaminating byproduct. Moreover, the utilization of a successive optimization technique, first over space and second over time, enables them to define the optimal intertemporal site specific deposit refund system. The final chapter in this area is by Fernandez, "Transboundary Water Management along the U.S.-Mexico Border". It presents an extensive survey of the studies concerning the problem of surface and groundwater resources shared by different nations. Special attention is given to the U.S.-Mexico border, where transboundary issues vary in terms of directions of flow and policy at national and binational levels. Typically, game theory concepts help examine incentives

XX

FRONTIERS IN WATER RESOURCE

ECONOMICS

between countries sharing water resources. There have been developments in policy structure from the historical to current context in which to assess formal channels of financing current and future water infrastructure along the border. The chapter also presents additional water analyses beyond the U.S.-Mexico border. Water quality (Chapters 9 -11): The chapter by Beare and Heaney, "Irrigation, Water Quality and Water Rights in the Murray Darling Basin, Australia", focuses on the development of water markets and the establishment of property rights to facilitate trade and create efficient incentives for investing in water use efficiency. However, irrigation can impose significant costs on both downstream water users and the environment. The extent to which trade and investments in improved water use technologies will lead to efficiency gains depends, in part, on how well water property rights account for the externalities associated with irrigation. The chapter presents an economic model of irrigation agriculture that is integrated with a hydrological model of the Murray-Darling River system to determine the potential magnitude of benefits of establishing property rights which take into account the downstream impacts of return flows. Establishing site specific conditions on property rights that partially internalize the external effects of irrigation may lead to an improvement in economic welfare. These could include taxes or subsidies on both trade between irrigation regions and investment in water saving technologies in specific irrigation regions. The next chapter by Khanna and Farnsworth, "Economic Analysis of Green Payments to Protect Water Quality", draws our attention to the economic literature on green payment policies. It provides an extensive review of both the theoretical and empirical literature on three basic issues in this field. What criteria should be used by a policy maker to select the recipients of these green payments? In a decentralized decision making situation, where participation in a green payment program is a voluntary decision of the farmer, how should these payments be designed to maximize social welfare or to achieve given environmental objectives cost-effectively? How efficient are green payment programs relative to first best instruments to control pollution that would have maximized social welfare? It shows the importance of incorporating spatial heterogeneity in the costs and benefits of alternative land uses while designing and targeting green payment policies. The final chapter in this area is by Roseta-Palma, "Conjunctive Use of Surface Water and Groundwater with Quality Considerations". It starts from the observation that deterministic models of conjunctive surface and groundwater management are not much more complicated than typical groundwater-only management models under simple assumptions. However, when water quality problems exist, the fact that there are two alternative sources of water gains a special significance, as there is no guarantee that they will be of comparable

INTRODUCTION

xxi

quality. Considering that water quality varies according to source, the paper analyzes some new implications that arise in a conjunctive system, with and without uncertainty in hydrological parameters. Production and technology adoption (Chapters 12 -13): The chapter by Hellegers, "The Impact of Recovering Irrigation Water Losses on the Choice of Irrigation Technology with Heterogeneous Land Quality and Different Crops", starts from the affirmation that an increase in the irrigation effectiveness - by adoption of modern irrigation technologies - is often proposed as a solution to water scarcity. Whether this is desirable in all cases is studied in this chapter. An existing conceptual farm model is used to show the impact of heterogeneous land quality on the choice of irrigation technology. Introduced in the model are recovery of irrigation water losses as well as the possibility of growing different crops on land with different quality, both extensions of existing work in this field. It becomes clear that recovery reduces resource costs -which makes adoption of modern irrigation technology less likely-, but may impose costs when recharge flows are of a bad quality -which makes adoption more likely. These costs are, however, often not paid by the farmer, which explains why adoption is sometimes subsidized. The final chapter of the book is by Zilberman, Cohen-Vogel and Reeves. "Precision Farming in Cotton" focuses on precision farming as a new category of technology that adjusts the application level of agricultural input and that accommodates variations within fields and also climatic and other seasonal variations. The authors develop a model of the impact of these technologies, recognizing heterogeneity in terms of utility and water holding capacity within fields. This is the first quantitative study of the adoption of varying input technology. Unlike the literature on adoption of irrigation technology, they recognize the importance of within-field heterogeneity and the capacity to adjust to heterogeneity. They develop general conditions under which precision technology will be adopted and show that introduction of precision technologies may lead to water saving and yield increasing effects in some cases while in others it may lead to water saving and yield reduction. They further develop a relationship that suggests that the impacts of the technology depend on the various dimensions of heterogeneity-within, using an example from California cotton fields and assuming uniform distribution of fertility and water holding capacity. They show that under plausible conditions technology adoption may result in a 10%-40% increase in profit.

Chapter 1 APPLICATION OF STOCHASTIC COOPERATIVE GAMES IN WATER RESOURCES* Ariel Dinar World Bank, U.S.A.

Stefano Moretti University of Genoa, Italy

Fioravante Patrone University of Genoa, Italy

Stefano Zara Via Roma, 30 - 09016 Iglesias (CA) - Italy

1.

INTRODUCTION

In March 2001, the Cahora Bassa (also known as Cabora Bassa) Dam on the Zambezi River in Mozambique could not hold the huge volume of water that filled its reservoir. As a result, devastating floods wiped out thousands of communities and their infrastructure, killing hundreds and leaving a quarter of a million others homeless. Zambia and Mozambique are riparians to the Zambezi, and the Cahora Bassa reservoir is filled with water flowing from Zambia. Dam operating rules that were set, taking into account average flow and hydropower preferences (over flood protection) are blamed for the disaster. The dam operating rule kept the *The authors thank an anonymous referee for very detailed and constructive comments.

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Dinar, Moretti, Patrone and Zara

water spillway at 80% of the average river flow, not leaving sufficient margin for extreme quantities. Was the damage (in both lives and economic loss) prevented, had the designers take into consideration extreme and stochastic events such that the 2001 flood? Situations similar to the Mozambique flood are very common around the world (e.g., the Elbefloodingof vast regions in the Czech Republic and Germany in Fall 2002). The problem of determining the operational level of a joint facility holds not only for dams. It is relevant also in setting the size and water quality in treatment facilities of sewerage and desalinization, and in any investment that could be affected by stochastic events. Thus, the issue of optimal design of a joint facility is of great economic importance. Game theory (GT) contributed to the understanding of allocation of costs and benefits of joint facilities among different users. In the water sector in particular, cooperative GT work (e.g., Gately, 1974; Loheman et al., 1979; Suzuki and Nakayama, 1976; Straffin and Heany, 1981; Dinar et al., 1986; Rogers, 1993; Moretti et al., 2002) addressed various aspects of water sector issues, including hydropower, water storage, multi-objective water projects, municipal sewage treatment, disposal and reuse, and international water cooperation. Most of the work, applying cooperative GT depart from the assumption of a deterministic world. As such, the proposed GT solutions to the allocation problems are probably restricted to a very narrow subset of average behavior of the events that affect the main input to the allocation problem, water. Dinar and Howitt (1997) attempted, in a very naive way, to address the problem of stochastic supply of water, affecting the design of a joint facility of drainage water treatment in the San Joaquin Valley in California. Although very specific to their problem. Dinar and Howitt identified that the stability of the allocation arrangement is sensitive to the state of nature, depending also on the selected allocation scheme. In view of the important role that water plays in regional and local projects, and taking into account that with climate change affecting the water cycle, the world is expected to face more stochastic and extreme events of water supply, incorporating stochastic consideration of water supply becomes more acute in designing water facilities. We shall provide a couple of examples (see Section 3) to show the effects of variability in water supply on the cooperative game used to model the cost allocation problem. Moreover, various water users may have different attitudes toward risk, depending on their economic, managerial, and institutional capacity. Therefore, the combination of stochastic events and players' risk attitude becomes increasingly an important issue in designing water related economic activities that depend on cooperation among the users. In this chapter we will apply a Stochastic GT framework, based on the work of Suijs and Borm (1999). Stochastic cooperative games have been applied in some fields (insurance: Suijs etal., 1998; network enterprises: Suijs, 2003; risk

Application of stochastic cooperative games in water resources trade for recovery projects after a disaster: Tanimoto et al., 2000). We consider that this analytical tool could be of use also in water-related problems, when variability in the input is high. We will use an example of a water treatment plant to illustrate the approach, although the principles hold also for cases of other water-related joint cost, such as for storage, etc. Our goal will be to offer, in a simplified setting, some examples of the problems which appear when one tries to take into account the stochastic aspects of the problem which is being modelled. We shall also show, by means of an example, that risk aversion of the players may influence the possibility of getting allocations which lie in the core of the game which models the cost allocation problem. Section 2 will introduce the general model, mainly describing the cost function that will be used. Section 3 describes cooperative games which arise under a couple of different interpretations of the cost function. Section 4 analyzes the core of a cooperative stochastic game, showing the relevance of risk aversion of the players. Section 5 concludes.

2.

THE GENERAL MODEL

Assume a finite set of possible states of nature: Q = {cji,... ^uj^}, associated with a probability distribution p{u;i). Assume also a cost function C : [0,+ooH [0,+oo[, about which we shall offer a couple of different interpretations. Assume further a given set of players (in this chapter we will use interchangeably the terms players and users) N — { 1 , . . . , n}. Each one of these players has to deal with some amount of polluted water, whose quantity depends also on the state of nature. For a given quantity of polluted water, the function C describes the cost for its treatment. In general, C will be derived by the application of an appropriate technology to the amount of water to be treated, and will depend also on a vector G of environmental variables such as water quality, soil properties, landscape, etc. For a given G, we assume that C — C{q) is obtained by choosing the most appropriate (less costly) technology. Notice that C can be affected by selecting the amount q of water to be applied and the technology in response to different states of nature (see for example Dinar and Zilberman, 1991; Dinar et al., 1992). Our general model does not allow for such responses at this stage. The main reason is that our model aims to address mainly ex-ante situations in various sectors. It can represent irrigation and drainage issues, municipal water supply and sewerage, hydropower and flood control, and industrial water use and sewerage. The model deals with ex-ante planning and management of the joint facility. For these two reasons it

3

4

Dinar, Moretti, Patrone and Zara

is assumed that both q and the technology t have to take into account the state of nature in the planning stage and cannot not be adjusted to it instantaneously. On the other hand, a different interpretation of the set of available technologies as the set of possible actions for the players of a stochastic cooperative game is provided in Section 4 (see Example). We assume also a function which describes for every player the quantity of polluted water that he has to deal with, for any given state of nature. A simplified version assumes that q{i^ u) = (/){i)q(u). This will be the case that we shall discuss in the Example. An almost obvious interpretation for this case is to assume that the quantity g(i,u;) of polluted water that each player has to deal with equals the product between the amount of water for unit area q{u) that reaches the soil (depending, trivially, on the weather conditions) and an idiosyncratic factor 0(i) representing the water to be treated (depending, for instance, on the amount, quality and characteristics of the soil and on the kind of cultivations). We are interested in showing how the players can jointly reduce the costs of treatment, and find reasonable ways in which they can divide the resulting joint costs; this procedure will be accomplished through the definition of an appropriate transfer utility game (TU-game). In a given state of nature cj G f^, the treatment cost for a given set of players 5 is given by C {Y^^^s ^(^^ ^)) • Moving to the aggregation on the states of nature, we suggest here, and develop in Section 3 (see considerations a) and b)), two different interpretations which can be of interest, each one of which leads to a TU-game. One interpretation is that c{S) - m a x C

Vg(i,a;) ) .

Another is that c(5) = X ] p ( u ; ) c ( ; ^ g ( ^ , u ; ) ] , iue^

\ieS

/

where p(a;), a; G O, is a probability distribution on the set of states of nature Q. In the case of a stochastic TU-game, a relevant factor is risk aversion of the players (more precisely, the different degrees of risk aversion that players may exhibit). In this chapter we focus on the case of risk neutral players; we provide just an illustrative example of a stochastic TU-game in which risk aversion of players, relaxing the stability conditions, can provide a non empty core, while risk indifference of the players would imply that the core is empty. In the case of risk neutral players, we use step cost functions as a basis for the cost calculations. These functions, which we study in Section 3, are not

Application of stochastic cooperative games in water resources concave, so that it will be no surprise that we can offer examples in which the core is empty or that the Shapley value lies outside the core. Step cost functions are clearly a simplified approach, but we believe that they contain one essential feature of the problem, with regard to stochastic factors: it can be assumed that the cost of water treatment is a concave function of the quantity of water use, but this realistically holds only for some range of the quantity. Whenever there is a strong variance in the water use input, it is difficult to land in a unique system of water treatment, which would give the concavity of the cost function. It is sound to use a given setup to treat water, but with some limits on the capacity, while for greater amounts of water to be treated it could be more appropriate to use other setups. Such setups could be more expensive in layout costs involved, or because they use a different technology. The alternative to using a setup that gives an answer to any quantity of polluted water, would mean to have an oversized treatment plant for almost all of the states of nature, and thus a much more expensive one, due to the inherent high fixed costs. In the following sections we detail these situations, providing numerical examples of the phenomena that we have described (empty core, Shapley value not in the core^).

3.

WATER GAMES

Water users have to tackle the ex-ante decision problem of how to share costs of water treatment (costs which are univocally realized only ex-post). What they precisely know is the probability distribution on occurrence of the states of nature and the cost function of treating water, which is depicted in Figure 1. We want to state clearly that we consider the simplest case of cost function that still exhibits interesting behavior from the viewpoint of cost allocation. It would be very easy to extend the analysis to more general cost functions (including more steps) than the one described in Figure 1 We argue that the ex-ante agreement between the players on the cost sharing rule to be adopted ex-post, is based just on the cost function and the probability distribution, and maybe also on the risk attitude of the players. So, it is very important to understand what is the meaning of a cost function such as the one depicted in Figure 1. We try to offer two different interpretations of the function in Figure 1 with regard to the case under consideration: a) A plant for water treatment has to be built. Solving this decision problem provides the cost function for the users. We assume that this decision problem will be solved according to the following criterion: if the amount of water to be treated is smaller than Q*, then a small plant should be built and the cost supplied is Ci; otherwise if the amount of water to be treated is larger than or equal to the amount Q*, then a bigger plant must

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be built and the cost is C2 (Ci < C2). Moreover, in this case we assume that for a given group of players S C N, if a state of nature a) s.t. YlieS^i^-'^) — Q* ^^ possible (i.e. the probability of d) to occur is larger than 0), then the group will decide to construct the larger plant and their cost is C2 (in fact, avoiding to build the bigger plant could be very dangerous, since users could face a situation of over-pollution). Otherwise, if for each state a; G fi with probability different than 0 the corresponding amount of polluted water q{u) is such that Ylies ^(^' ^) ^ Q*' ^^^^ ^he group will build the smaller plant and the cost will be Ci; b) A plant for water treatment has already been built. Then for each state of nature u E ft and for any group of players S such that J2ieS ^(^' ^) ^ Q*, the cost of water treatment in the given plant is Ci. If a group of players S has to treat a quantity Y2ieS ^(^' ^) — Q* ^^ some states of nature u e Q, the cost becomes C2 in such states of nature. In this interpretation, the increase in cost could be due to some facilities/services needed only when the amount Q* has been exceeded (for example the renting of tankers used to transport liquid to another place). Note that these are "extreme" cases that will be modelled using the treatment cost functions defined in Section 2, c and c respectively. Our analysis could be extended to other situations in the sense that we could be able to provide

cost of water treatment (in $)

quantity of water

Figure 1. Cost function of water treatment

Application of stochastic cooperative games in water resources examples analogous to those we have just introduced to treat more general functions as in Figure 2. In this figure it is shown a case in which the cost function continuously increases with the quantity of water that is affected by different states of nature (e.g. due to an increment of rainfall events which implies a consequent growth in terms of facilities/services for water treatment) and again there is a quantity Q* that, if reached, requires the ex-ante construction of a new plant (with corresponding cost increases).

cost of water treatment (in $) C2

Ci

Q* Figure 2.

quantity of water

An intermediate situation between interpretation (a) and (b)

In the following sections, we will refer to a cost function like the one depicted in Figure 1, in order to define two cooperative games, following the interpretations (a) and (b), respectively.

3.1

Considerations on interpretation (a)

Consider the case with 3 players N =^ {1,2,3} and a set of 3 states of nature Suppose that q(i^uj) = (f){i)q{co), where (^(1) = (f){2) = 1, (l){3) — 2 (reflecting three farms with the land for player 3 being twice as big as for players 1 and 2), and g(cc;^) = Qi, describes the amount of water to be treated. For each state a; G fi and for each player 1, 2, 3, values are reported in the following table: state of the world \ players 1 2 3

Ui

UJ2

W3

Qi Qi 2Qi

Q2 Q2 2Q2

Qi

Qz 2Q3

We furthermore assume that Qi = 2, Q2 == 3, Q3 = 4 and Q* == 10.9 (here Q* refers to Figure 1). Moreover we suppose that there is a given probability distribution on the states cji, ct;2, c^s different from zero for each sate. Notice

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Dinar, Moretti, Patrone and Zara

that in this interpretation the quantification of the probability attached to the states of nature is not relevant, as long as it can be assumed positive for each of them. Then, by interpretation (a), the characteristic cost function values for each state of nature for the resulting cooperative game of sharing the joint water treatment cost are summarized in the following schema:

OJl

{1}

{2} {3} {1,2} {1.3} {2,3} {1,2,3}

Ci Ci Ci

Ci Cl Ci Ci

Wz Ci

Ci

Ci

Ci

Ci Ci Ci Ci C2

Ci Ci Ci

WS

C2 C2

max cost Ci Ci Ci Ci Ci C2 C2

For example, consider coalition {2,3}: the cost for the plant under state uj2 is Ci since g(2,cc;2) + g(3,c Q*. Therefore, coalition {2,3} could face an amount of polluted water greater than Q* (i.e. state CJ3). Therefore, the cost for coalition {2, 3} in the resulting (ex-ante) game should be the maximum cost C2. We are going to consider the resulting (ex-ante) game ({1, 2, 3}, c), where c(0) = 0, c({l}) - Ci, c({2}) = Ci, c({3}) = Ci, c({l,2}) = Ci, c({l, 3}) - Ci, c({2,3}) = C2 and c({l, 2,3}) = C2. This game is not concave. In fact a cost game (A^, c) is concave if for each i G {1, 2, 3} and each S, T such that 5 C T C AT \ {i} c{T U {i}) - c(T) < c{S U {i}) - c{S),

(1)

which is not satisfied for coalitions S* == {1} and T = {1,2} and i = 3. The core Core{{l^ 2,3}, c) of the game ({1, 2,3}, c) is given by all the vectors {xi,X2->xz) e'M? such that: V i e {1,2,3} 3^1 + X2 < C i ^1 + 3:3 < C i a^l + ^2 + 3:3 = ^^2

Then, we can write:

Application of stochastic cooperative games in water resources •

Xi 2Ci, since in such a case the core is empty. But we are left with the interesting case that for 2Ci > C2 > | C i the Shapley value does not lie in the core, even if it is not empty. This is the case, for example, if Ci = 11 and C2 = 20.

3.2

Considerations on interpretation (b)

Following interpretation (b), we are going here and in Section 4 to perform calculations on the ex-ante game deduced by the probability distribution on the

Application of stochastic cooperative games in water resources

11

states of nature and, as consequence, on the related costs for water treatment. Of course, we must keep in mind that ex-ante costs could reasonably be very different than ex-post, since they depend on the occurrence ex-post of a particular state of nature. More precisely, under the current interpretation of the cost function in Figure 1.1, the theoretical considerations on the ex-ante cooperative game are only aimed to find a ("fair") cost sharing agreement of the cost incurred by the larger coalition ex-post. Consider 3 players N — {1,2,3} and a set of 3 states of nature fi = For each state uj ^Q. and for each player i e N, values of g(z, LU) representing the amount of water to be treated are reported in the following table: state of nature players 1 2 3

CJi

UJ2

CJS

Qi Oi 2Qi

Q2

Qs Qs

O2 2O2

2Q3

where Qi = 2, Q2 = 3, Q3 = 5 and Q* = 10.9. Moreover we suppose a probability distribution on Q such that prob{(jJi) = | , proh{u)2) =• | and proh{ujz) = ^. Then, by interpretation (b), the characteristic cost function for each state of nature is summarized in the following schema:

{1} {2} {3} {1,2} {1,3} {2,3} {1,2,3}

Ul

ljJ2

W3

Ci Ci Ci Ci Ci Ci Ci

Ci Ci Ci Cl Ci Ci

Ci Ci Ci Ci

C2

C2 C2 C2

average cost Ci Ci Ci Ci f Ci + iC72 = Ci + i(C2 - Ci) iC7i + iC2 = Ci + i ( C 2 - C i ) i C i + iC2 = Ci + | ( C 2 - C i ) 1

For example, consider coalition {2, 3}: the cost for the treatment plant under state UJ2 is Ci since g(2, ^2) + ^(3, UJ2) = Q2 + 2Q2 = 9 < Q*, whereas under stateu;3thecostshouldbeC2sinceg(2,C(;3)+g(3,u;3) = (53+2^3 = 15 > Q*. Therefore, the cost for coalition {2,3} in the resulting (ex-ante) game should be the average cost reported in the previous table. Now we are going to consider the resulting (ex-ante) game ({l,2,3},c), where c{S) is given by the average cost of 5 C A^ as in the previous table. Also the average game is not concave for C2 > Ci. In fact, the condition (1) is not satisfied by coalitions 5 == {1} and T = {1,3} and i = 2. The core Core{{l^ 2, 3}, c) of the game ({1,2,3}, c) is given by all the vectors (xi, X2j X3) G M.^ such that:

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Xi 1{C2 - Cl) 1(^2 - Cl) ^ 9Ci > 4C2 ^ Ci> |C2. So if C2 > f Cl then Core{{l, 2, 3}, c) is empty. Is the Shapley value in C6>r^({l, 2, 3}, c)? Consider the following table concerning the incremental contribution of players 1,2, 3 to the cost of each permutation in the game ({1,2,3}, c): 123 132 213 231 312 321 total i =

1 Cl Cl 0

i(C2-Ci) t(C2-Ci) t(C2-Cl) 2(7i + |(C2 - Cl) ^ + ^(C2-Cl)

2 0

^(C2-Cl) Cl Cl

3

i(C2-Ci) ^(C2-Cl) f(C2-Ci) KC2 - Cl)

Cl i(C2-Ci) ^(C2 - Cl) Cl 1 2 C i + J ( C 2 - C i ) _ 2Ci +_|(C2 - Ci)J -3^ + ^(C'z - C l ) ^ + ^ ( C 2 - C l ) 1

Application of stochastic cooperative games in water resources For individual rationality of players 1,2,3: ^1 - 02 - f + ^ ( C 2 - C i ) < Ci ^ ^ ( C 2 - C i ) < fCi ^ 5(C2-Ci) < 20Ci, h = M ( C 2 - Ci) < | C i ^ 8(C2 - Ci) < 20Ci. For coalitional rationality for coalitions {1,3} and {2, 3} c/^i + h = cl>2 + h = f Ci + i ( C 2 - Ci) < Ci + ^(C2 - Ci) ^ ^ ^ ( C 2 - Ci) < ^Ci 4:^ 14(C2 - Ci) < 20Ci and for coalition {1,2} (/>! + 02 = fCi + |§(C2 - Ci) < Ci ^ ^ ( C 2 - Ci) < i C i 4^ 4:^ 20(C2 - Ci) < 20Ci ^ C2< 2Ci. Note that if the last constraint is satisfied then also the other constraints are satisfied. Therefore, if C2 > 2Ci thentheShapley valueisnotinC6>r^({l, 2, 3},c). So far our game does not take into account the attitude of the players toward risk. Will the results, and especially the core conditions change with the introduction of risk attitude of the players?

4.

AN EXAMPLE OF CALCULATION OF THE CORE FOR A STOCHASTIC COOPERATIVE GAME

The model considered in this section is aimed to deal with the risk attitude of the players. First, a preliminary introduction to stochastic cooperative games (scg) (Suijs, 1998; Suijs et al., 1999; Suijs and Borm, 1999). An scg is a tuple T = \^N, {AS)SCNJ

{XS)SCN,

{hi)ieN) where

• ^ 5 is the (finite) set of possible actions which coalition S C N can take. • A stochastic variable Xs{a) corresponds to each action a G As, that is: Xs ' As -^ L^(R), with L^(R) being the set of stochastic variables with finite expected value. •

{t:i)ieN is the preference profile for players in N on stochastic payoff (cost).

An allocation of Xs{a) to the players in S is represented by a pair with Y^i^s ^i — 0' YlieS n == 1 and n > 0 for all players i e S. Given such a pair (d, r|a), agent i e S receives the stochastic cost di + riXs{a). The second part, riXs{a), describes the fraction of Xs{a) that is allocated to player i. The first part, di, describes the deterministic transfer payments between the players^. When di < 0 then player i receives money while di > 0 means that this player pays money. The purpose of these transfer payments is that the players compensate each other for transfers of random costs. For example, a risk-adverse player (that "hates" uncertainty) who receives a large fraction of Xs(a) can be compensated by the other players if they give him an adequate

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negative amount di. The set Z(S), S C N, of allocations that coalition S can obtain contains all such allocations for all Xs{a). Note the relevance of indicating which action has been taken in correspondence to a given allocation. Having in mind the standard framework of a deterministic TU-game, one could object that allocations can be represented simply as repartitions of a cost (or a gain) and the actions performed do not take a relevant role in the modelling process. In the stochastic case, instead, actions are essential, as shown in the following very elementary example, where the set of possible actions represents the set of possible technologies for water treatment for players. Example Consider a group of two players 1 and 2, that can choose between only two possible actions (i.e. technologies which can independently be used for a complete water treatment): action ti means "treating water by technology one" and action ^2 means "treating water by technology two". Moreover, suppose that, for budget constraints, only one technology can be implemented by the players (e.g. initial fixed costs of implementation are too high for using both of them). Of course the cost of water treatment is conditioned, for both of the available technologies, on the state of nature which is going to occur. Figure 3 shows the cost functions related to the two technologies ti and ^2, when employed to treating a certain amount of water.

k cost of water treatment (in $)

technology

ti^-^

^^^^^.^--^^echnology t\

\m^^

1

1 1 \

\